Hypergeometric Distribution Calculator — drawing without replacement

A population of N items contains K “successes”. You draw n items without putting them back. How likely is it that exactly k of them are successes? Pick an example or enter your own numbers.

Example

An example only fills in the numbers — the formula is always the same. Changing a number switches to “Your own values”.

Population size (N)

All items you draw from, e.g. 52 cards or 49 lottery numbers.

Successes in the population (K)

Items that count as a success, e.g. 4 aces, or the 6 numbers that will be drawn.

Sample size (n)

How many items are drawn without replacement.

Successes in the sample (k)

The number of successes you ask about. X can only take values from max(0, n − (N − K)) to min(n, K).

P(X = k): 1.77%

The same as: —
P(X ≤ k): —
P(X ≥ k): —
Expected value E(X): —
Variance Var(X): —
Standard deviation σ: —
Possible values of X: —
In words: —

Full distribution of X

Every possible value of X with P(X = k), P(X ≤ k) and P(X ≥ k)
k	P(X = k)	P(X ≤ k)	P(X ≥ k)
Fill in the fields above — the result is computed on this page.

The P(X = k) column adds up to 100%. P(X ≤ k) grows and P(X ≥ k) falls from row to row; the highlighted row is your k.

What do these numbers mean?

Result for the settings above: 1.77%. It updates as soon as any field changes.
We compute it from the formula shown below — not from a prediction model and not from draw history.
The main result is for exactly k successes. “At most” sums from zero through k; “at least” sums from k through the possible maximum.
Urn example: N = 10, K = 4, n = 3, k = 2 gives 30%. Sampling is without replacement.

The population is finite and sampling is without replacement. The binomial distribution requires independent trials with constant probability; it is not always interchangeable with this model.

How do I use this page?

Fill in the fields in the form above — the result recomputes at once, with no page reload.
Under the result you will find the interpretation scale, if the calculator has one: it says which range your value fell into.
The formula we use is shown lower on this page — you can check the arithmetic yourself.
To compute the same for another game or variant, change it in the select field — prices and pools update automatically.

What this page does not do: The population is finite and sampling is without replacement. The binomial distribution requires independent trials with constant probability; it is not always interchangeable with this model.

Formula

P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n)   ·   E(X) = n × K/N   ·   Var(X) = n × K/N × (1 − K/N) × (N − n)/(N − 1)

N is the population size, K the number of successes in it, n the sample size and k the number of successes in the sample. C(a, b) is the binomial coefficient. X can only take values from max(0, n − (N − K)) to min(n, K). For N = 1 (or n = N) the variance is 0 — the result of the draw is then certain.

Theory — what is the hypergeometric distribution?

The hypergeometric distribution describes drawing without replacement from a population of N items, of which K are “successes”. Each item removed changes the composition of what remains, so the chance of success in the next draw depends on earlier outcomes. This is exactly how balls in a lottery drum, cards dealt from a deck, or a quality-control sample behave. It differs from the binomial distribution, which assumes sampling with replacement and a constant success probability on every trial.

Frequently asked questions

What is the difference between the hypergeometric and the binomial distribution?

The binomial distribution assumes drawing with replacement, so the chance of success stays the same every time. The hypergeometric distribution is for drawing without replacement, where each draw changes what is left. For a large population and a small sample the two give similar results.

Why is P(X ≥ k) not equal to 1 − P(X ≤ k)?

Because both include the row X = k. The correct relation is P(X ≥ k) = 1 − P(X ≤ k − 1), which you can check in the table.

Hypergeometric Distribution Calculator — drawing without replacement

Full distribution of X

Formula

Read more

Frequently asked questions

See also

Ancient Egyptian units converter

Arithmetic Mean

Basic calculator

My saved selections

Notifications after draws